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A2-Level Maths: Statistics 2for Edexcel

S2.1 Binomial and Poisson distributions

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Binomial distributions

Binomial distributions

Mean and variance of a binomial

Use of binomial tables

The Poisson distribution

Poisson tables

Mean and variance

Approximating a binomial by a Poisson

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Many real-life situations can be modelled using statistical distributions. Examples of the types of problem that can be addressed using these distributions include:

Special distributions

In a board game, players needs a six before they can start. What is the probability that they haven’t started after 5 tries?

What proportion of the adult population have an IQ above 120?

The number of accidents on a stretch of motorway averages 1 every 2 days. How likely is it that there will be no accidents in a week?

12% of people are left-handed. What is the probability that a class of 30 people will have more than 6 left-handed people?

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Binomial distribution

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Introductory example:A spinner is divided into four equal sized sections marked 1, 2, 3, 4.

If the spinner is spun 6 times, how likely is it to land on 1 on four occasions?

The number of possible sequences is

Binomial distribution

!

! !6

4

6

4 2C

!. .

!..

!4 26

0 25 0 754 2

0 0330

One possible sequence would be 1 1 1 1 1′ 1′.

(i.e. the number of ways of arranging 6 items, where 4 are of one kind and 2 are of a different kind).

Each sequence has probability 0.254 × 0.752.

So the required probability is

Most calculators have an nCr

button

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A binomial distribution arises when the following conditions are met:

Binomial distribution

If the above conditions are satisfied and X is the random variable for the number of successes, then X has a binomial distribution. We write: X ~ B(n , p).

an experiment is repeated a fixed number (n) of times (i.e., there is a fixed number of trials);

the outcomes from the trials are independent of one another;

each trial has two possible outcomes (referred to as success and failure);

the probability of a success (p) is constant.

n and p are called parameters.

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Which of these situations might reasonably be modelled by a binomial distribution?

Binomial

Not binomial

Binomial

Binomial distribution

Joan takes a multiple choice examination consisting of 40 questions. X is the number of questions answered correctly if she chooses each answer completely at random.

A bag contains 6 blue and 8 green counters. James randomly picks 5 counters from the bag without replacement. X is the number of blue counters picked out.

A bag contains 6 blue and 8 green counters. Jan randomly picks 5 counters from the bag, replacing each counter before picking the next. X is the number of blue counters picked out.

Outcomes are not independent

1

2

3

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Which of these situations might reasonably be modelled by a binomial distribution?

Notbinomial

Not binomial

Binomial

Binomial distribution

Jon throws a dice repeatedly until he obtains a six. X is the number of throws he needs before a six arises.

Judy counts the number of silver cars that pass her along a busy stretch of road. X is the number of silver cars that pass in a minute.

Josh is a mid-wife. He delivers 10 babies. X is the number of babies that are girls.

1

2

3

The number of trials is not fixed

The number of trials is not fixed

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Binomial distribution

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If X ~ B(n , p), then

where q = 1 – p.

Binomial distribution

( ) , , ,...P for 0 1 2n x n xxX x C p q x n

( ) . . .12 1 111P 1 0 4 0 6 0 01741X C

( ) . . .12 3 93P 3 0 4 0 6 0 142X C

Number of possible

sequences

Probability of x

successes

Probability of n – x failures

( ) . . . .12 0 12 120P 0 0 4 0 6 0 6 0 00218X C

So P(X > 1) = 0.980 (3 s.f.)

a) (to 3 s.f.)

b) P(X > 1) = 1 – P(X = 0) – P(X = 1).

Example: X ~ B(12, 0.4).

Find a) P(X = 3)b) P(X > 1).

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Binomial distribution

( ) . . .10 4 64P 4 0 51 0 49 0 197X C a)

b)

( ) ( ) ( ) ( )

. . . .

.

.

. . .

10 8 2 10 9 108 9

P 8 P 8 P 9 P 10

0 51 0 49 0 51 0 49 0 51

0 04945 0 01144 0 0011

0 21

9

06

X X X X

C C

Example: The probability that a baby is born a boy is 0.51. A mid-wife delivers 10 babies. Find:

a) the probability that exactly 4 are male;

b) the probability that at least 8 are male.